Azimuth and Quadrant Bearing Conversion
The round trip the two COGO tiles point at but the declination tile never made: an azimuth (0 to 360 deg clockwise from north, the form all the coordinate math runs on) and a quadrant bearing (N or S, an angle 0 to 90 deg, E or W, the form deeds and plats are written in) name the same direction. First quadrant reads N az E; second, S the supplement E; third, S the excess over 180 W; fourth, N the deficit from 360 W. An azimuth of 138.5 deg is S 41 deg 30 min 00 sec E; SW 30 deg is azimuth 210. The cardinal azimuths 0/90/180/270 read as due north/east/south/west. Shown to the nearest second. Plane-direction bookkeeping only: no magnetic declination (that is the bearing-conversion tile) and no grid convergence.
Formula and source
an azimuth A (0 to 360 deg clockwise from north) maps to a quadrant bearing by quadrant: A in [0,90] is N A E; A in [90,180] is S (180 minus A) E; A in [180,270] is S (A minus 180) W; A in [270,360] is N (360 minus A) W. The reverse runs each rule backward: N b E gives A = b, S b E gives A = 180 minus b, S b W gives A = 180 plus b, N b W gives A = 360 minus b. The cardinal azimuths 0/90/180/270 read as due north/east/south/west.
Azimuth and quadrant (deed) bearing are the two standard ways to write a plane direction in US surveying references (Ghilani/Wolf, Elementary Surveying; FM 5-233, public-domain), by name; the conversion is exact plane geometry.
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